Skip to main content
SpringerLink
Log in

Direct and inverse secondary resonance effects in the spherical motion of an asymmetric rigid body with moving masses

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

Secondary resonance effects in the spherical motion of a heavy asymmetrical rigid body with moving masses are reviewed in the case close to the Lagrange top. It is known that perturbations acting on the body, particularly a small displacement of the center of mass with respect to axis of symmetry and a small perturbing moment in a connected system of coordinates, lead to non-resonant evolutions of angular velocity. These evolutionary phenomena are called direct secondary resonance effects. They are visible in a dynamic system consisting of an asymmetric rigid body and several masses fixed relative to the body. However, the presence of moving masses connected to a rigid body by springs may lead to the stabilization of the angular velocity of the body. This dynamic phenomenon should be attributed to the inverse secondary resonance effect. The aim of this paper is to study the characteristic direct and inverse secondary resonance effects in a perturbed spherical motion of a rigid body with fixed and moving masses. The method of integral manifolds and the averaging method are used for the asymptotic analysis of the secondary resonance effects. The paper presents the numerical results of modeling the direct and inverse resonance effects.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Chernousko, F.L., Akulenko, L.D., Leshchenko, D.D.: Evolution of Motions of a Rigid Body About its Center of Mass. Springer, Cham (2017)

    Book  Google Scholar 

  2. Akulenko, L.D., Leshchenko, D.D., Rachinskaya, A.L.: Quasi-optimal deceleration of rotational motion of an asymmetric body in resistive medium. J. Comp. Syst. Sci. Int. 53(3), 338–344 (2014)

    Article  Google Scholar 

  3. Akulenko, L., Leshchenko, D., Kushpil, T., Timoshenko, I.: Problems of evolution of rotations of a rigid body under the action of perturbing moment. Multibody Syst. Dyn. 6(1), 3–16 (2001)

    Article  Google Scholar 

  4. Neishtadt, A.I.: Evolution of rotation of a solid acted upon by the sum of a constant and a dissipative perturbing moment. Mech. Solids 15(6), 21–27 (1980)

    Google Scholar 

  5. Zabolotnov, YuM: The resonance motions of a statically stable Lagrange top at small nutation angles. J. Appl. Math. Mech. 80(4), 302–310 (2016)

    Article  MathSciNet  Google Scholar 

  6. Shabana, A.A.: Dynamics of Multibody Systems. Cambridge University Press, Cambridge (2013)

    Book  Google Scholar 

  7. Neishtadt, A.I., Vasiliev, A.A., Artemyev, A.V.: Capture into resonance and escape from it in a forced nonlinear pendulum. Regul. Chaot. Dyn. 18(6), 686–696 (2013)

    Article  MathSciNet  Google Scholar 

  8. Sadov, Y.A.: Rapid rotation of the satellite with a magnetic damper. 4 Dissipative Evolution. Resonance Effects. Cosm. Res. 24(4), 553–564 (1986)

  9. Sadov, Y.A.: Secondary resonance effects in mechanical systems. Mech. Solids 25(4), 20–24 (1990)

    Google Scholar 

  10. Sanders, J.A., Verhulst, F., Murdock, J.: Averaging Methods in Nonlinear Dynamical Systems. Springer, New York (2007)

    MATH  Google Scholar 

  11. Zabolotnov, Y.M., Lyubimov, V.V.: Non-linear resonance evolutionary effects in the motion of a rigid body about a fixed point. J. Appl. Math. Mech. 66(3), 401–408 (2002)

    Article  MathSciNet  Google Scholar 

  12. Zabolotnov, Y.M., Lyubimov, V.V.: Secondary resonance effects in the rotation of a rigid body about a fixed point. Mech. Solids 37(1), 39–46 (2002)

    MATH  Google Scholar 

  13. Lyubimov, V.V.: External stability of a resonance in a nonlinear system with slowly changing variables. Mech. Solids 37(6), 42–47 (2002)

    Google Scholar 

  14. El-Gohary, A.: Global stability of rigid body containing moving masses. Int. J. Nonlinear Mech. 36(4), 663–669 (2001)

    Article  MathSciNet  Google Scholar 

  15. El-Gohary, A.: On the orientation of a rigid body using point masses. Appl. Mech. Comput. 151(1), 163–179 (2004)

    MathSciNet  MATH  Google Scholar 

  16. Zabolotnov, YuM, Lyubimov, V.V.: Application of the method of integral manifolds for construction of resonant curves for the problem of spacecraft entry into the atmosphere. Cosm. Res. 41(5), 453–459 (2003)

    Article  Google Scholar 

  17. Shchepakina, E.A., Sobolev, V.A., Mortell, M.P.: Singular Perturbations. Introduction to System Order Reduction Methods with Applications. Lecture Notes in Mathematics. Springer, Cham (2014)

    MATH  Google Scholar 

  18. Lyubimov, V.V.: Asymptotic analysis of the secondary resonance effects in the rotation of a spacecraft with small asymmetry in the atmosphere. Russ. Aeronaut. 57(3), 245–252 (2014)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Samara University, 34, Moskovskoye Shosse, Samara, Russia, 443086

    Vladislav V. Lyubimov

Authors
  1. Vladislav V. Lyubimov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladislav V. Lyubimov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lyubimov, V.V. Direct and inverse secondary resonance effects in the spherical motion of an asymmetric rigid body with moving masses. Acta Mech 231, 4933–4946 (2020). https://doi.org/10.1007/s00707-020-02790-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-020-02790-9

Search

Navigation